Preheat oven to 350 degrees F (175 degrees C).

Beat white sugar and 2/3 cup softened butter together in a bowl using an electric mixer until smooth and creamy; mix in 1/3 cup milk, egg, and 1 1/2 teaspoons vanilla extract. Stir flour, 1/2 cup cocoa powder, baking powder, and salt into creamed butter mixture until dough is just combined; fold in 1 cup candy-coated peanut butter pieces. Drop dough, 1 to 2 tablespoons per cookie, onto a baking sheet.

Bake in the preheated oven until edges of cookies are crisp, 10 to 12 minutes. Allow cookies to cool on pans for 2 to 3 minutes before transferring to wire racks to cool completely.

Beat 3 tablespoons melted butter and 3 tablespoons cocoa powder together in a bowl using an electric mixer until smooth and creamy; beat in confectioners' sugar, 2 tablespoons milk, and 1/2 teaspoon vanilla extract until frosting is fluffy and smooth. Frost cookies and decorate with more candy-coated peanut butter pieces.